Resistors remain an important part of integrated circuits (ICs). However, presently available circuit simulators do not provide significant support for resistors. Since there is no international standard for resistor modeling, IC manufacturers have developed individual resistor models and have included such models into the simulators as modules, subcircuits or subroutines. The accuracy of such approaches depends on the model itself and upon how well it is matched to measured data. For example, resistors are generally modeled for low electric field (less than 1×105 V/m) applications because of the ease at which such measurements are made, and because low-voltage measurement systems are readily available. For low voltages or electric fields, a commonly used model resistor equation is given by the following:
            R      =                        R          SH                ⁢                              L            eff                                W            eff                          ⁢                  vco          ·          tco                      ,    where              vco      =              (                  1          +                      VCR            ⁢                                                  ⁢                          1              ·                              (                                                      Δ                    ⁢                                                                                  ⁢                    V                                                        L                    eff                                                  )                                              +                      VCR            ⁢                                                  ⁢                          2              ·                                                (                                                            Δ                      ⁢                                                                                          ⁢                      V                                                              L                      eff                                                        )                                2                                                    )              ,                  or        ⁢                                  ⁢        vco            =              (                  1          +                      VCR            ⁢                                                  ⁢                          1              ·              Δ                        ⁢                                                  ⁢            V                    +                      VCR            ⁢                                                  ⁢                          2              ·              Δ                        ⁢                                                  ⁢                          V              2                                      )              ,    and              tco      =              (                  1          +                      TCR            ⁢                                                  ⁢                          1              ·                              (                                  T                  -                                      T                    nom                                                  )                                              +                      TCR            ⁢                                                  ⁢                          2              ·                                                (                                      T                    -                                          T                      nom                                                        )                                2                                                    )              ,  where RSH is the Sheet Resistance; Leff is the effective length of the resistor; Weff is the effective width of the resistor; VCR1 is the 1st-order coefficient for bias effects; VCR2 is the: 2nd-order coefficient for bias effects; ΔV is the voltage difference between the high and low terminals of the resistor; TCR1 is the 1st-order coefficient for temperature effects; TCR2 is the 2nd-order coefficient for temperature effects; T is the global temperature of the resistor (that is, the junction temperature at thermal equilibrium); Tnom is a reference temperature, 25° C. being used for most cases; vco is the voltage coefficient term; and tco is the temperature coefficient term. Note that the second definition provided for vco requires a redefinition of VCR1 and VCR2 as including 1/Leff and (1/Leff)2, respectively.
Polynomial expressions for both vco and tco have been used for some time because of ease of implementation in the circuit simulator software packages yielding rapid simulation results and mathematical convergence. Variations of these expressions have been used to reflect special situations specific to individual users, but multiplication of a voltage coefficient term, a temperature coefficient term, and a zero-bias resistance term is standard.
The function of a resistor model is to accurately characterize current as a function of voltage (I-V) applied to the resistor. As may be observed in FIG. 1 hereinbelow, the existing approach for modeling the resistance as a function of 2nd order polynomial of bias voltage (or electric field) fails for high electric fields. The conventional polynomial voltage coefficient model does not yield physically meaningful results because it predicts a negative differential resistance characteristic at high electric fields. By contrast, the measured I-V characteristics for a resistor at high electric fields are similar to those of metal oxide field effect transistors in that the current saturates above a certain electric field due to velocity saturation (mobility degradation due to high lateral electric fields) of the carriers. A possible cause of this incorrect prediction may be the use of 2nd order polynomial functions since, as the electric field increases, the predicted resistance value is dominated by the square of the electric field, and the observed saturating current behavior is not predicted.
Accordingly, it is an object of the present invention to provide a method for characterizing resistor behavior at high applied electric fields.
Another object of the present invention is to provide a method for characterizing resistor behavior at high applied electric fields using measurements obtained at low applied electric fields.
Additional objects, advantages and novel features of the invention will be set forth in part in the description that follows, and in part will become apparent to those skilled in the art upon examination of the following or may be learned by practice of the invention. The objects and advantages of the invention may be realized and attained by means of the instrumentalities and combinations particularly pointed out in the appended claims.